A min-max characterization of Zoll Riemannian metrics
Differential Geometry
2022-05-03 v2 Algebraic Topology
Geometric Topology
Abstract
We characterize the Zoll Riemannian metrics on a given simply connected spin closed manifold as those Riemannian metrics for which two suitable min-max values in a finite dimensional loop space coincide. We also show that on odd dimensional Riemannian spheres, when certain pairs of min-max values in the loop space coincide, every point lies on a closed geodesic.
Keywords
Cite
@article{arxiv.1809.08689,
title = {A min-max characterization of Zoll Riemannian metrics},
author = {Marco Mazzucchelli and Stefan Suhr},
journal= {arXiv preprint arXiv:1809.08689},
year = {2022}
}
Comments
24 pages, 1 figure; version 2: Theorem 1.1 now stated for manifolds homeomorphic to odd-dimensional spheres (the original statement was correct, but indeed the assumptions made on the manifold forced it to be a topological sphere according to Bott-Samelson theorem and the Poincar\'e conjecture); minor correction in the centered equation before (5.5)